Sums of Twin Primes: Multiples of 6 and 12 (April 8 and 9)

Last week we explored again the Sieve of Eratosthenes using a grid of 6 columns instead of the traditional ten. The children made an amazing discovery: every prime number except for 2 and 3 is one more than a multiple of 6 or one less than a multiple of 6. One of the Mathletes remarked how if you added two consecutive primes you would get a multiple of 6 and another in the same group immediately noticed that they also are multiples of 12. They are talking about the amazing properties of Twin Primes. These are two prime numbers separated by 2.

For centuries, mathematicians have studied twin primes including the discovery that there is an infinite number of these twins. However, currently the largest known twin prime pair is 3756801695685 · 2⁶⁶⁶⁶⁶⁹ ± 1. It has 200,700 digits.

The first and second graders proved that each sum of a twin prime above 3 and 5 was a multiple of 6. They did this by first adding a list of 80 twin primes (it is critical that the children start by adding the ones column first; if that sum is greater than 9, they should write down the units digit of the ones column of that sum and carry the tens digit [a one] above the next column). Then they had to confirm that each sum was both even (sums that end in 0,2,4,6,or 8) AND that the sum of the digits in the twin prime sum is a multiple of 3. If both of these conditions are true, the sum is a multiple of 6. For example, if we add the twin primes 1,301 and 1,303 we get a sum of 2,604. Clearly it is divisible by 2 because it ends in an even number, 4. In addition, when we add the digits of the sum 2+6+0+4=12; and 12 is a multiple of 3 so 2,604 is a multiple of 3. Since the multiples 2 and 3 have a product (2 times 3) of 6, we know that 2,604 is a multiple of 6. The attached pdf for this lesson also contains an answer key as well as the first 171 twin primes among the first 1,000 prime numbers. So the children an continue this quest with larger primes like 7,877 + 7,879.

The third graders and beyond were challenged with proving that each of the sums of the twin primes above 3 and 5 are also multiples of 12. The test for multiplicity of 12 is a little more challenging but very attainable. The sum must be a multiple of 3 and 4. We have already discussed the test for multiplicity of 3 above by adding the digits of the number (this test only works for multiples of 3 and 9). The test for multiplicity of 4 is to look at the last two digits of the sum (for example, if we are looking at the twin primes of 2,657 and 2,659 and get a sum of 5,316, we simply look at the last two digits of 5,316, to see that 16 is divisible by 4), and if it is divisible by 4, so is the whole number.

Most of these groups will also be learning my short division method of dividing by 12. A best practice is to write down the first 9 multiples of 12 (12,24,36,48,60,72,84,96,108). I have listed the steps for this process on page 5 of the attached pdf called Twin Primes Multiples of 12. I also included the first 171 twin primes and last page of a 250 page document listing the first 100,000 twin primes (what is important to note about this page is that it only lists the first of the twin primes, so to find its twin you have to add two to the number). For example, The 100,000th twin prime is listed as 18,409,199, but to get the actual second number in the twin, you have to add two to this number to get 18,409,201. And yes, if you add them together you get 36,818,400. If you divide this by 12, you will get 3,068,200 without a remainder.

I would love to see all of the adding and division work that the children complete. Please encourage them to write the division problem in a larger font and spread the number so there is room for the remainders.